182 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
We have then 
u- a amir mer 
dy 
a eels D) dU av om) 19 ¥ 
v= 24 42/2 Th ae aaa a + + $2 1D ee (89) 
we 3 re _ 
atl \dz dy 
The ordinary approximate theory obtains differential equations to determine 
U,V. These are easily found by eliminating E and K in turn from (88). 
Thus 
Cau dV = 2 2 
“re ‘ae “a 
aU aV_ 
-—8 v-K 
dy da (ay ) 
and 
tad Oi aN, aU GNEN eas 
a+l1 a dx dy )+az( dy ~ da ls 16 uh (90) 
te Ce wv) 1 a av) = Ni 
atl dy\idu dy Sax dy dx  16uh 
The principal parts of the contribution of Z force to extensional displacements 
appear in (72), (78). In the notation of those formulee 
d _» 
VA 
5 aa 
v= VE a—3 2! 
| : 32rph > 
= “yi 
Q 
w= 
Qa 
with in addition, w = the odd part in z of the ambiguous term in (78). 
If these last values of u, v be included in the principal values U, V, then the right- 
hand members of (90) will become respectively 
1 a-—3 dZ 1 a—3 aZ 
Te 2 aia) F Teun -¥ + aaa 35 | ; : : ; - (91) 
28. Approximate values of the stresses across a plane parallel to the faces. 
For any distribution of force parallel to the faces of the plate, the formulee of — 
§§ 26, 27 give the terms of the two lowest orders in the values of wu, v, and the term’ 
of lowest order in w.* From these terms we can calculate all the stresses but % to a 
first approximation, and as in § 22, when the first term of 7 is known, we can find 
two terms of az, x, and yy. This first term of % we may get very easily from the 
symbolical form of the solution ae to (80), (81). Thus for areal force X 
eS Gs Oey 5 
w= $3(2-2) 7 + gya2 (Wz - ae ee Ua : f ; eC) 
* There should be added from (84) the terms 
w= #F(z-2')X/2u, v= (2-2)Y/2Qu, w=0. 
